Year 11 Maths: Variance and Standard Deviation

Introduction to Variance and Standard Deviation

Hello, Year 11! Today, we’re going to explore two important concepts in statistics: variance and standard deviation. These tools help us understand how data spreads out or varies. Let’s break it down step by step.

What is Variance?

Variance measures how far a set of numbers is spread out from their average (mean). If the numbers are all close to the mean, the variance is small. If the numbers are far apart, the variance is large.

How to Calculate Variance:

1. Find the mean (average) of the numbers.
2. Subtract the mean from each number to find the deviation.
3. Square each deviation (to make it positive).
4. Find the average of these squared deviations.

The formula for variance (

$$

\sigma^2 = \frac{\sum (x_i – \mu)^2}{N}

$$

Where:

• $$x_i$$ = each value in the data set
• $$\mu$$ = mean of the data set
• $$N$$ = number of values in the data set

What is Standard Deviation?

Standard deviation is simply the square root of the variance. It gives us a measure of spread in the same units as the data. A smaller standard deviation means that the data points are closer to the mean, while a larger one means they are more spread out.

The formula for standard deviation (

$$

\sigma = \sqrt{\sigma^2}

$$

Example

Let’s say we have the following data set: 2, 4, 4, 4, 5, 5, 7, 9.

1. Calculate the mean: $$\mu = \frac{2 + 4 + 4 + 4 + 5 + 5 + 7 + 9}{8} = \frac{40}{8} = 5$$
2. Calculate the deviations from the mean:
   • 2 – 5 = -3
   • 4 – 5 = -1
   • 4 – 5 = -1
   • 4 – 5 = -1
   • 5 – 5 = 0
   • 5 – 5 = 0
   • 7 – 5 = 2
   • 9 – 5 = 4
3. Square the deviations:
   • (-3)² = 9
   • (-1)² = 1
   • (-1)² = 1
   • (-1)² = 1
   • (0)² = 0
   • (0)² = 0
   • (2)² = 4
   • (4)² = 16
4. Find the average of these squared deviations: $$\sigma^2 = \frac{9 + 1 + 1 + 1 + 0 + 0 + 4 + 16}{8} = \frac{32}{8} = 4$$
5. Calculate the standard deviation: $$\sigma = \sqrt{4} = 2$$

So, the variance is 4, and the standard deviation is 2.

Key Rules and Tips

• Always square the deviations when calculating variance.
• Standard deviation is easier to interpret since it is in the same unit as your data.
• A low variance or standard deviation indicates that the data points are close to the mean, while a high one indicates a wider spread.

Questions

Easy Level Questions

1. Find the variance of the data set: 1, 2, 3.
2. Calculate the mean of: 5, 10, 15.
3. What is the standard deviation of: 2, 2, 2?
4. Find the variance of: 4, 4, 4, 4.
5. If the mean is 3 and the data is 1, 2, 4, 5, what is the variance?
6. What is the standard deviation of: 7, 7, 7, 7, 7?
7. Find the mean of: 10, 20, 30, 40.
8. If the numbers are 8, 8, 10, what is the variance?
9. What is the standard deviation of: 0, 0, 0?
10. Calculate the variance of: 2, 3, 4, 5.

Medium Level Questions

1. Find the variance of: 1, 3, 5, 7, 9.
2. Calculate the mean of: 2, 4, 6, 8, 10.
3. What is the standard deviation of: 3, 7, 7, 9?
4. Find the variance of: 10, 20, 30, 40, 50.
5. If the mean is 6 and the data is 5, 7, 9, what is the variance?
6. Calculate the standard deviation of: 1, 1, 2, 2, 3, 3.
7. Find the mean of: 12, 15, 18.
8. If the numbers are 4, 4, 6, 10, what is the variance?
9. What is the standard deviation of: 1, 2, 3, 4, 5, 6?
10. Calculate the variance of: 5, 5, 10, 15.

Hard Level Questions

1. Find the variance of: 10, 20, 30, 40, 50, 60.
2. Calculate the standard deviation of: 1, 1, 1, 10, 10, 10.
3. What is the variance of the data set: 2, 4, 4, 4, 5, 5, 7, 9?
4. Find the mean of: 100, 200, 300, 400.
5. If the mean is 50 and the data is 40, 60, 70, what is the variance?
6. Calculate the variance of: 5, 6, 7, 8, 9, 10.
7. What is the standard deviation of: 3, 5, 8, 8, 10?
8. Find the variance of: 1, 2, 3, 4, 5, 100.
9. Calculate the mean of: 6, 7, 8, 9, 10.
10. If the numbers are 15, 20, 25, 30, 35, what is the standard deviation?

Answers

Easy Level Answers

1. Variance = 1
2. Mean = 10
3. Standard Deviation = 0
4. Variance = 0
5. Variance = 2
6. Standard Deviation = 0
7. Mean = 25
8. Variance = 1
9. Standard Deviation = 0
10. Variance = 1.25

Medium Level Answers

1. Variance = 8
2. Mean = 6
3. Standard Deviation = 2
4. Variance = 250
5. Variance = 4
6. Standard Deviation ≈ 0.82
7. Mean = 15
8. Variance = 4
9. Standard Deviation ≈ 1.71
10. Variance = 12.5

Hard Level Answers

1. Variance = 350
2. Standard Deviation ≈ 3.67
3. Variance = 4
4. Mean = 250
5. Variance = 100
6. Variance = 2.5
7. Standard Deviation = 2.24
8. Variance = 1,210
9. Mean = 8
10. Standard Deviation = 7.07

I hope this helps you understand variance and standard deviation better! Don’t hesitate to ask questions if you’re unsure about anything!